Topics Covered Monday, Aug 29, 05: Section 1.4 of Straight: Existential and universal quantifiers and their negation, negation of p -> q, contarpositive of p -> q, examples on writing a statement as a formula. Tuesday, Aug 30, 05: Section 1.6 of Straight (Proof techniques): direct proofs, direct proof of p -> q, proof by a counterexample (to show that a statement of the form "forall x in S, p(x)" or "forall x in S, p(x) -> q(x)" is false. Wednesday, Aug 31, 05: Continuation of 1.6, indirect proofs (by contrapositive and by contradiction), proof by cases, proof of "if and only if", proof of "the following are equivalent". Friday, Sep 2, 05: Continuation. Tuesday, Sep 6, 05: Continuation. Wednesday, Sep 7, 05: Section 1.7 of Straight (Mathematical Induction). Friday, Sep 9, 05: Continuation and strong form of mathematical induction. Monday, Sep 12, 05: resolution proofs and start Chapter 2 of Straight (sections 2.1 - principle of well-ordering). Tuesday, Sep 13, 05: Section 2.2 (the division algorithm), finding the quotient and the remainder, related theorems. Wednesday, Sep 14, 05: quiz, section 2.3 (the Eucildean algorithm), finding the least common multiple; finding the greatest common divisor and writing it as a linear combination of the two numbers; relatively prime. Friday, Sep 16, 05: continuation. Monday, Sep 18, 05: Section 2.4 (prime numbers), how to determine efficiently if a number is prime or composite, finding the smallest prime factor, factoring a positive integer as a product of primes and writing that in the canonical form. Tuesday, Sep 20, 05: Finish Section 2.4, Section 2.5 (integers modulo n), adding and multiplying in Zn, additive and multiplicative inverses, when the multiplicative inverse exists and how to find it. Wednesday, Sep 21, 05: solving equations in Zn, Friday, Sep 23, 05: concluding remarks of Chapter 2, define matrices and vectors and introduce some notation. Monday, Sep 26, 05: Addition and multiplication of matrices, dot product of vectors. Tuesday, Sep 27, 05: Exam I. Wednesday, Sep 28, 05: Friday, Sep 30, 05: Special matrices (diagonal, upper-triangular, lower-triangular, zero matrix, identity matrix), definition of inverse. Monday, Oct 3, 05: Tuesday, Oct 4, 05: Transpose and powers of matrices; theorems. Wednesday, Oct 5, 05: Cofactors, determinants. Friday, Oct 7, 05: Inverses. Monday, Oct 10, 05: Continue with inverses, using the inverse to solve a linear system of equations in which the matrix of coefficients is invertible. Tuesday, Oct 11, 05: Begin Chapter 2 of Goodaire, sets, sequences, and multisets, definitions. Monday, Oct 17, 05: Finish Section 2.2 (sets and operations on them). Tuesday, Oct 18, 05: Section 2.3 (binary relations), symmetric, reflexive, transitive, antisymmetric, and equivalence relations, equivalence classes. Wednesday, Oct 19, 05: continuation. Friday, Oct 21, 05: more examples, disjpint, pairwise disjoint, partitions, equivalence classes, comparible elements, total orders. Monday, Oct 24, 05: quiz, examples of partial orders and total orders, posets. Tuesday, Oct 25, 05: representation of a binary relation as a digraphs and as a matrix, composition of binary relations, inverse of a binary relation, more examples. Wednesday, Oct 26, 05: Begin Chapter 3 of Goodaire: functions, definition, functions and binary relations, domain, range, target, image, preimage, one-to-one (injective). Friday, Oct 28, 05: one-to-one continued, onto (surjective), one-to-one correspondence (bijection). Monday, Oct 31, 05: previous topics continued. Tuesday, Nov 1, 05: Quiz, Inverse. Wednesday, Nov 2, 05: Inverse continued, composition of functions, related theorems. Friday, Nov 4, 05: cardinality, countably infinite, countable. Monday, Nov 7, 05: Finish Chapter 3 of Goodaire. Tuesday, Nov 8, 05: Begin Chapter 6 of Goodaire: Inclusion and exclusion. Wednesday, Nov 9, 05: Addition and multiplication rules (Section 6.2) Friday, Nov 11, 05: permutations (Section 7.1). Monday, Nov 14, 05: solution of homework and other exercises. Tuesday, Nov 15, 05: Exam II (Material: Chapter 3 of Straight and the related material we covered, chapters 2 and 3 of Goodaire and the related material we covered. Wednesday, Nov 16, 05: combinations (Section 7.2). Friday, Nov 18, 05: Repetitions (generalized combinations and permutations) (Section 7.3). Monday, Nov 28, 05: Section 7.3 continued, derangements(Section 7.4). Tuesday, Nov 29, 05: Section 7.4 continued, the Pigeon-Hole Principle (Section 6.3). Wednesday, Nov 30, 05: Section 6.3 continued, the Binomial Theorem (Section 7.5). Friday, Dec 2, 05: Finish 7.5. Monday, Dec 5, 05: Define linear transformations, theorem. Tuesday, Dec 6, 05: Kernel and range of a linear transformation, examples, and theorem. Wednesday, Dec 7, 05: the matrix of a linear transformation, examples.